Nes̆etřil's conjecture on the chromatic bound in the GIP theorem

Let $F$ be a graph with at least one edge. The cited theorem provides a constant $c(F)$ such that, for every natural number $r$, there is a graph $G$ with $\chi(G)>r$, $\omega(G)=\omega(F)$, and every induced subgraph of $G$ without an induced copy of $F$ having chromatic number at most $c(F)$. Nes̆etřil's conjecture. Theorem GIP holds with

$$c(F)=\chi(F).$$

The source says this conjecture would give a positive answer to Davies's question, but does not state a resolution.